Why Matching Attained Ages Matters
A select mortality table describes rates for lives recently selected at a particular age. The rates can vary with both the age at selection and the time since selection. An ultimate table, by contrast, gives rates by attained age without separating lives by how recently they were selected. To compare the two tables fairly, compare rates for the same attained age, not merely entries appearing in the same row or column.
For instance, a life selected at age 50 and observed one year later has attained age 51. The appropriate ultimate comparison is therefore the ultimate rate at age 51. Comparing that select rate with the ultimate rate at age 50 would mix different ages and could give a misleading impression of the effect of selection.
The bracketed age identifies the age at selection; the duration after the plus sign identifies how long the life has been under observation since selection. In particular, \(q_{[x]+t}\) and \(q_{x+t}\) refer to the same attained age but need not be equal. If the select period lasts \(n\) years, select rates apply for durations \(0,1,\ldots,n-1\); after that period, the table uses ultimate rates.
Measuring the Difference
There are several useful ways to report a matched-age comparison. It is important to identify which one is being used: the signed difference indicates direction, while the absolute difference does not.
A negative signed difference means the select rate is lower; a positive one means it is higher. The relative reduction is a proportion of the ultimate rate, not a proportion of the select rate. It is meaningful as a reduction only when the select rate is lower. Selection often produces lower rates soon after selection, with the difference diminishing over later durations, but this is a possible pattern, not a rule that every table must satisfy.
| Quantity | Expression | Interpretation |
|---|---|---|
| Signed difference | \(q_{[x]+t}-q_{x+t}\) | Direction and size of the select-minus-ultimate difference |
| Absolute difference | \(\left|q_{[x]+t}-q_{x+t}\right|\) | Size of the difference, without direction |
| Relative reduction, if select is lower | \(\dfrac{q_{x+t}-q_{[x]+t}}{q_{x+t}}\) | Fraction of the ultimate rate by which the select rate is lower |
Worked Example: Comparing Rates Across a Three-Year Select Period
A hypothetical three-year select table gives the following annual rates for a life selected at age 40: \(q_{[40]}=0.010\), \(q_{[40]+1}=0.012\), and \(q_{[40]+2}=0.015\). The corresponding ultimate rates are \(q_{40}=0.014\), \(q_{41}=0.015\), and \(q_{42}=0.017\). Compare the rates at each duration, including the relative reduction when applicable.
Method. Match each select rate with the ultimate rate at the same attained age. The three attained ages are 40, 41, and 42, respectively.
At duration 0, the signed difference is \(0.010-0.014=-0.004\). The absolute difference is \(0.004\), and the relative reduction is \[ \frac{0.014-0.010}{0.014}=\frac{0.004}{0.014}=0.285714\ldots, \] or \(28.57\%\).
At duration 1, the signed difference is \(0.012-0.015=-0.003\). The absolute difference is \(0.003\), and the relative reduction is \[ \frac{0.015-0.012}{0.015}=\frac{0.003}{0.015}=0.20, \] or \(20.00\%\).
At duration 2, the signed difference is \(0.015-0.017=-0.002\). The absolute difference is \(0.002\), and the relative reduction is \[ \frac{0.017-0.015}{0.017}=\frac{0.002}{0.017}=0.117647\ldots, \] or \(11.76\%\).
Check and interpretation. Each comparison uses matching attained ages, and all three select rates are lower than their ultimate counterparts. The relative reductions decrease from \(28.57\%\) to \(11.76\%\), illustrating a diminishing difference across these durations. The common mistake avoided here is comparing \(q_{[40]+1}\) with \(q_{40}\), even though the former applies at attained age 41.
How the Rate Comparison Extends Across Several Years
A one-year rate comparison does not by itself give the probability of death over several years. For a life who remains subject to the select basis during the period, each annual survival probability is \(1-q_{[x]+t}\). Multiplying these conditional survival probabilities gives the probability of surviving all the years. The same reasoning applies to ultimate rates.
The justification is the multiplication rule for conditional probabilities. To survive \(k\) years, the life must survive year 1, then survive year 2 conditional on surviving year 1, and continue in this way. If the conditional survival probabilities are \(1-q_{[x]},1-q_{[x]+1],\ldots\), the chain rule gives their product. The probability of death within the \(k\)-year period is the complement of surviving the entire period.
Here \({}_kp_{[x]}\) and \({}_kq_{[x]}\) denote, respectively, the probability of surviving and the probability of dying within \(k\) years for a life selected at age \(x\). If a calculation crosses the end of the select period, use the select rates only for the durations in that period and use the appropriate ultimate rates thereafter.
Worked Example: Comparing Three-Year Death Probabilities
A hypothetical select basis for a life selected at age 30 has annual death rates \(0.020\), \(0.023\), and \(0.027\) at durations 0, 1, and 2. The ultimate rates at the matching attained ages 30, 31, and 32 are \(0.024\), \(0.027\), and \(0.030\). Calculate and compare the three-year death probabilities.
Method. Calculate each three-year survival probability as the product of the three successive annual survival probabilities, then subtract from 1.
On the select basis, the survival probability is \[ (1-0.020)(1-0.023)(1-0.027) =0.980(0.977)(0.973). \] The first two factors give \(0.980(0.977)=0.95746\). Therefore, \[ 0.95746(0.973)=0.95746-0.95746(0.027) =0.95746-0.02585142=0.93160858. \] The select-basis three-year death probability is \(1-0.93160858=0.06839142\), or \(0.0684\) to four decimal places.
On the ultimate basis, the survival probability is \[ (1-0.024)(1-0.027)(1-0.030) =0.976(0.973)(0.970). \] The first two factors give \(0.976(0.973)=0.949648\). Therefore, \[ 0.949648(0.970)=0.92115856. \] The ultimate-basis three-year death probability is \(1-0.92115856=0.07884144\), or \(0.0788\) to four decimal places.
The ultimate-basis probability exceeds the select-basis probability by \[ 0.07884144-0.06839142=0.01045002, \] which is \(0.0105\), or \(1.0450\) percentage points, to four decimal places. Check: the select death probability is smaller, consistent with each of the three matched select rates being lower. The common mistake avoided is multiplying the wrong annual survival factors or treating a three-year death probability as the sum of the annual rates.
Reading Selection Effects Across Durations
A select table can be examined duration by duration by comparing each select rate with the ultimate rate at the same attained age. This separates two influences: attained age and duration since selection. A rate may rise with duration simply because the attained age rises; that alone does not show that the effect of selection is weakening. To assess the selection pattern, compare the select and ultimate rates at each attained age, then examine how the differences or relative reductions change across durations.
For example, a smaller relative reduction at duration 2 than at duration 0 suggests that the select rate is closer to the ultimate rate by duration 2. It does not prove that every later select rate must move closer, nor does it establish why the rates differ. A table gives mortality assumptions; conclusions should be limited to the comparisons supported by its entries.
For \(q_{[x]+t}\), record selection age \(x\) and duration \(t\).
Use \(x+t\); this is the age to which the annual select rate applies.
Compare with \(q_{x+t}\), not with the ultimate rate at the age of selection unless \(t=0\).
Report a signed or absolute difference as requested; use a relative reduction only when the select rate is lower.
Worked Example: Detecting an Age-Indexing Trap
A hypothetical table gives \(q_{[50]+1}=0.010\), \(q_{50}=0.011\), and \(q_{51}=0.012\). Determine the correctly matched ultimate rate and calculate the relative reduction. Then identify the result from the incorrect age match.
Method. The life was selected at age 50 and has completed one year, so the select rate applies at attained age \(50+1=51\). The correct ultimate comparison is \(q_{51}=0.012\).
The signed difference is \(0.010-0.012=-0.002\), and the absolute difference is \(0.002\). Since the select rate is lower, the relative reduction is \[ \frac{0.012-0.010}{0.012} =\frac{0.002}{0.012} =0.166666\ldots, \] or \(16.67\%\).
The incorrect match uses \(q_{50}=0.011\). That would produce a relative reduction of \((0.011-0.010)/0.011=0.090909\ldots\), or \(9.09\%\), but it compares a rate at attained age 51 with one at age 50. It is therefore not the requested matched-age comparison. Check: adding one year to the selection age gives 51, confirming the correct ultimate index.
Common Mistakes and Practical Interpretation
- Matching on the wrong age. A select rate at duration \(t\) is matched with ultimate age \(x+t\), not automatically with the selection age \(x\).
- Calling a signed difference absolute. \(q_{[x]+t}-q_{x+t}\) can be negative. The absolute difference is \(\left|q_{[x]+t}-q_{x+t}\right|\).
- Using the wrong denominator for relative reduction. The reduction relative to ultimate mortality is divided by \(q_{x+t}\), not by \(q_{[x]+t}\).
- Assuming select rates must always be lower. A table may have a select rate equal to or greater than its matched ultimate rate. In that case, do not describe the comparison as a reduction.
- Adding annual death probabilities for a multi-year probability. Successive years are conditional on survival to those years. Multiply the annual survival probabilities and take the complement.
These comparisons are useful for checking that a table has been read correctly and for summarizing how its select and ultimate assumptions differ. They also provide a direct way to interpret the selection pattern without confusing it with the effect of changing attained age. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, match rates at the same attained age and show enough work to distinguish a rate comparison from a multi-year probability calculation.
- A hypothetical table has \(q_{[46]+2}=0.018\) and \(q_{48}=0.020\). Find the signed difference, absolute difference, and relative reduction. (Answer: signed difference \(-0.002\); absolute difference \(0.002\); relative reduction \(10.00\%\).)
- A life selected at age 35 has select rates \(0.012\) and \(0.016\) for the next two years. The matching ultimate rates are \(0.015\) and \(0.019\). Calculate both two-year death probabilities and their difference, ultimate minus select. Round probabilities to four decimal places. (Answer: select \(0.0278\); ultimate \(0.0337\); difference \(0.0059\).)
- For a life selected at age 60, the select rates at durations 0, 1, and 2 are \(0.025\), \(0.030\), and \(0.036\). The ultimate rates at ages 60, 61, and 62 are \(0.028\), \(0.032\), and \(0.038\). Find the relative reduction at durations 0 and 2, and state whether it declines. (Answer: duration 0, \(10.71\%\); duration 2, \(5.26\%\); yes, it declines.)
- A table gives \(q_{[70]+1}=0.041\), \(q_{70}=0.039\), and \(q_{71}=0.045\). Which ultimate rate is correctly matched, and what is the relative reduction? (Answer: \(q_{71}=0.045\); relative reduction \((0.045-0.041)/0.045=8.89\%\).)
- At one duration, a matched select rate is \(0.031\) and the ultimate rate is \(0.029\). Give the signed and absolute differences, and state whether a relative reduction applies. (Answer: signed difference \(0.002\); absolute difference \(0.002\); no relative reduction applies because the select rate is higher.)